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Theorem grpoidinv 27692
Description: A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1 𝑋 = ran 𝐺
Assertion
Ref Expression
grpoidinv (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
Distinct variable groups:   𝑥,𝑦,𝑢,𝐺   𝑢,𝑋,𝑥,𝑦

Proof of Theorem grpoidinv
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 474 . . . . . . . 8 (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → (𝑢𝐺𝑧) = 𝑧)
21ralimi 3090 . . . . . . 7 (∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
3 oveq2 6822 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑢𝐺𝑧) = (𝑢𝐺𝑥))
4 id 22 . . . . . . . . 9 (𝑧 = 𝑥𝑧 = 𝑥)
53, 4eqeq12d 2775 . . . . . . . 8 (𝑧 = 𝑥 → ((𝑢𝐺𝑧) = 𝑧 ↔ (𝑢𝐺𝑥) = 𝑥))
65rspccva 3448 . . . . . . 7 ((∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
72, 6sylan 489 . . . . . 6 ((∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
87adantll 752 . . . . 5 (((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
98adantll 752 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑢𝐺𝑥) = 𝑥)
10 simpl 474 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → 𝐺 ∈ GrpOp)
1110anim1i 593 . . . . . 6 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝐺 ∈ GrpOp ∧ 𝑥𝑋))
12 id 22 . . . . . . . . . 10 ((𝐺 ∈ GrpOp ∧ 𝑢𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢𝑋))
1312adantrr 755 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → (𝐺 ∈ GrpOp ∧ 𝑢𝑋))
1413adantr 472 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝐺 ∈ GrpOp ∧ 𝑢𝑋))
152adantl 473 . . . . . . . . 9 ((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
1615ad2antlr 765 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
17 simpr 479 . . . . . . . . . . 11 (((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
1817ralimi 3090 . . . . . . . . . 10 (∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
1918adantl 473 . . . . . . . . 9 ((𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2019ad2antlr 765 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2114, 16, 20jca32 559 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ((𝐺 ∈ GrpOp ∧ 𝑢𝑋) ∧ (∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)))
22 grpfo.1 . . . . . . . 8 𝑋 = ran 𝐺
23 biid 251 . . . . . . . 8 (∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧 ↔ ∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧)
24 biid 251 . . . . . . . 8 (∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢 ↔ ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)
2522, 23, 24grpoidinvlem3 27690 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝑢𝑋) ∧ (∀𝑧𝑋 (𝑢𝐺𝑧) = 𝑧 ∧ ∀𝑧𝑋𝑤𝑋 (𝑤𝐺𝑧) = 𝑢)) ∧ 𝑥𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
2621, 25sylancom 704 . . . . . 6 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢))
2722grpoidinvlem4 27691 . . . . . 6 (((𝐺 ∈ GrpOp ∧ 𝑥𝑋) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
2811, 26, 27syl2anc 696 . . . . 5 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑥𝐺𝑢) = (𝑢𝐺𝑥))
2928, 9eqtrd 2794 . . . 4 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (𝑥𝐺𝑢) = 𝑥)
309, 29, 26jca31 558 . . 3 (((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) ∧ 𝑥𝑋) → (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
3130ralrimiva 3104 . 2 ((𝐺 ∈ GrpOp ∧ (𝑢𝑋 ∧ ∀𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))) → ∀𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
3222grpolidinv 27685 . 2 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑧𝑋 ((𝑢𝐺𝑧) = 𝑧 ∧ ∃𝑤𝑋 (𝑤𝐺𝑧) = 𝑢))
3331, 32reximddv 3156 1 (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  ran crn 5267  (class class class)co 6814  GrpOpcgr 27673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-ov 6817  df-grpo 27677
This theorem is referenced by:  grpoideu  27693  grpoidval  27697  grpoidinv2  27699  grpomndo  34005
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